Evaluation of Acid Treatments from Pressure Build-Up Analysis

Influence of Shape and Skin of Matrix-Rock Blocks on Pressure Transients in Fractured Reservoirs

Summary

A formulation for pressure transients in terms of the intrinsic, or core, properties of the two media that compose the fractured reservoir establishes the influence of these properties on-and their corroboration fromthe pressure/time relationship observed in well and interference tests. The influence of the following reservoir characteristics are analyzed: the area of fractures transverse to flow; the dimensions, shape, and properties of rectangular parallelepiped matrix-rock blocks; and permeability reduction in the block surfaces. permeability reduction in the block surfaces. A restatement of the so-called quasisteady-state intermedia flow provides Warren and Root's a and parameters with the physical meaning provides Warren and Root's a and ? parameters with the physical meaning they lacked and allows direct determination of the blocks' minimum dimensions.

Introduction

To describe pressure transients in naturally fractured reservoirs, Warren and Root assumed that flow occurs only in the fractures, and that the volume of the fractures is distributed in the whole bulk volume. In contrast, de Swaan describes the problem in the net volume of the fractures. Furthermore, Warren and Root's approach includes an approximate interaction between matrix-rock blocks and fractures, whereas de Swaan's theory describes the transient flow of a slightly compressible fluid between both media in a totally causal way that allows intuitive insight into the peculiar shape of the pressure/time plot observed in well tests. In this paper, the intuitive pressure/time plot observed in well tests. In this paper, the intuitive aspect is extended further to account for the effects of shape and surface damage of the matrix-rock blocks on pressure transients.

Carslaw and Jaeger's approach is embraced in the development of the present work: "It is possible to express the differential equation and boundary conditions in terms of dimensionless variables... This procedure is attractive from the pure mathematical point of view and makes the mathematics a little more concise. It will not be adopted here since the physical significance of a formula is clearer if it is expressed in the original physical variables."

Theory

Volumetric Relationships in the Fractured Medium. Assuming that the reservoir is composed of uniformly distributed matrix-rock blocks (hereafter called "blocks") and fractures associates the volume of every block with half the volume of its surrounding fracture (Fig. 1). Thus, the reservoir is an aggregate of elementary or repetitive cells. No orthogonality of the fractures is implied in this simplification when the blocks are proposed to be rectangular parallelepipeds because this is only an approximation to the actual parallelepipeds because this is only an approximation to the actual shape of the blocks typical in a reservoir region.

Decline Curve Analysis Using Type Curves for Two-Porosity Systems

Abstract

Constant producing pressure solutions that define declining production rates with time for a naturally fractured reservoir are presented. The solutions for the dimensionless flow rate are based on a model presented by Warren and Root. The model was extended to include constant producing pressure in both infinite and finite systems. The results obtained for a finite no-flow outer boundary are new and surprising. It was found that the flow rate shows a rapid decline initially, becomes nearly constant for a period, and then a final decline in rat,- takes place.A striking result of the present study is that ignoring the presence of a constant flow rate period in a type-curve match can lead to erroneous estimates of the dimensionless outer radius of a reservoir. An example is presented to illustrate the method of type-curve matching for a naturally fractured system.

Introduction

Naturally fractured reservoirs consist of heterogeneous porous media where the openings (fissures and fractures) vary considerably in size. Fractures and openings of large size form vugs and interconnected channel, whereas the tine cracks form block systems which are the main body of the reservoir (Fig. 1). The porous blocks store most of the fluid in the reservoir and are often of low permeability, whereas the fractures have a low storage capacity and high permeability. Most of the fluid flow will occur through the fissures with the blocks acting as fluid sources. Even though the volumetric average permeability in a naturally fractured system is low, such systems often exhibit an effective permeability that is higher than the block matrix permeability, and behave differently from ordinary homogeneous media.

These systems have been studied extensively in the petroleum literature. One of the first such studies was published by Pirson in 1953. In 1959, Pollard presented one of the first pressure transient models available for interpretation of well test data from two-porosity systems. The most complete analysis of transient flow in two-porosity systems was presented in 1960 by Barenblatt and Zheltov. The Warren and Root study in 1963 is considered the forerunner of modern interpretation of two-porosity systems. Their paper has been the subject of study by many authors. The behavior of fractured systems has long been a topic of controversy Many authors have indicated that the graphical technique proposed by Pollard in 1959 is susceptible to error caused by approximations in the mathematical model. Nevertheless, the Pollard method still is used. The most complete study of two-porosity systems appears to be the Mavor and Cinco-Ley study in 1979. This study considers wellbore storage and skin effect, and also considers production, both at constant rate and at constant pressure. However, little information is presented concerning the effect of the size of the system on pressure buildup behavior.Although decline curve analysis is widely used, methods specific to two-porosity fractured systems do not appear to be available. It is the objective of this paper to produce and study decline curve analysis for a naturally fractured reservoir. The Warren and Root model was chosen as the basis for this work.

Partial Differential Equations

The basic partial differential equations for fluid flow in a two-porosity system were presented by Warren and Root in 1963. The model was extended by Mavor and Cinco-Ley to include wellbore storage and skin effect.

SPEJ

P. 354^